It doesn't hurt to validate the domain part, and make sure it has at least one MX or A record. With a bit of AJAX you can usually validate that before the user even submits the form. Not forgetting to take punycode into account.
The trouble with validating using SMTP is that it is fairly likely to introduce a delay. I still think it is better to just check the DNS, with a very low timeout and then send the email as long as there wasn't a negative response.
You can let the user immediately know that the email was sent, but then you can also push updates to the user whilst they're still on the site if there is a rejection/bounce.
I wrote https://emailprivacytester.com/, which does the DNS checks that I mentioned when you enter your email address, and then keeps you informed of the status of the message delivery as it happens, including any SMTP rejection messages.
Those methods are far from foolproof. Lots of sites that do greylisting for example, will give a temporary failure before the message even reaches the server that actually knows what addresses on the domain are valid.
Also none of the e-mail systems I've operated in the last 15 years or so will let on whether or not the user actually exists until at earliest when you have committed to sending a message, and many of them wouldn't even then (instead accepting the message and sending a bounce) to reduce the spam harvesting.
